Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{4r^2 - 60r + 224}{-5r^2 - 10r + 315}$
First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {4(r^2 - 15r + 56)} {-5(r^2 + 2r - 63)} $ $ z = -\dfrac{4}{5} \cdot \dfrac{r^2 - 15r + 56}{r^2 + 2r - 63} $ Next factor the numerator and denominator. $ z = - \dfrac{4}{5} \cdot \dfrac{(r - 7)(r - 8)}{(r - 7)(r + 9)}$ Assuming $r \neq 7$ , we can cancel the $r - 7$ $ z = - \dfrac{4}{5} \cdot \dfrac{r - 8}{r + 9}$ Therefore: $ z = \dfrac{ -4(r - 8)}{ 5(r + 9)}$, $r \neq 7$